[FOM] Freiling's darts, Martin's Axiom, and CH

A decade ago Solomon Feferman posed a key foundational question -- What are
the axioms of set theory (i.e. ZFC) axioms *for*? (in his "Does Mathematics
Need New Axioms?"). Six decades before that Kurt Godel had hopes of giving
what would have been an answer, with his constructible universe of sets L.
Since that time sentiment has been largely against the axiom (V=L), with
most (including Godel!) coming to believe that the universe V of sets must
include more than just the constructible sets.
Harvey Friedman has said ZFC is the "gold standard" of axioms for set
theory, and of course that's right, *even though* there is still not
consensus on exactly what it is we are talking about when we speak of sets!
But, so what? Aestheticians may well have their own gold standard for
characteristics of Beauty without having any consensus on what exactly
Beauty is.
In this posting I want to present an analogy: simple groups, groups, and
semigroups corresponding to the constructible universe, other types of
universe, and still other. Note to start with that simple groups are a
nontrivial class of groups, though not comprising the "usual and customary"
objects of study in a first college course; also, that a number of things
that can be said about groups *remain correct assertions* about semigroups,
a more encompassing class of entities than groups (but again, not the
usual and customary objects of study).
The Generalized Continuum Hypothesis (GCH) is true in L, but is there an
axiom plausible (in terms of intuition, consequences, or both) that gives us
"almost" the simple results of the CH, but actually allows for ~CH? Of
course, Martin's Axiom (MA) does the job. MA(k) is a statement about
partial orders (or about open dense sets) having a certain desirable
property for a family of sets such that the cardinality of the family is
not more than k; MA is the axiom that MA(k) holds for all cardinals less
than c (i.e. 2^{aleph_0}).
Here is a topological version of MA: In every compact Hausdorff space
satisfying the countable chain condition (ccc), the intersection of fewer
than continuum many open dense sets is not empty. (A topological space X
satisfies ccc if each disjoint family of open subsets of X is countable.)
Some seemingly "nice" consequences of MA (among many others): All subsets of
the Reals of cardinality < c must have Lebesgue measure zero (D.A. Martin
and R. Solovay); any model of ZFC + GCH has a generic extension in which MA
holds, and also 2^{aleph_0} = aleph_2 (S. Shelah); and a strengthening of MA
called the proper forcing axiom also gives that 2^{aleph_0} = aleph_2.
(These last results are coherent with results of W. Easton that let us
conclude "for all regular cardinals k, 2^{aleph_k} = {aleph_k + 2}" is
relatively consistent with ZFC.)
Now for the group analogy. Models of ZFC which answer to MA *perhaps* are
like groups in terms of usual and customary results within the models; and
they *may* tell us something about (yet not be candidates for) the ultimate
structure (the role played by semigroups in our analogy).
In particular, taking another look at Chris Freiling's "Axiom of Symmetry"
I propose not to argue for it here as an axiom candidate, rather to think
of it as the "method of Freiling's darts" or some such thing. (It is well
documented on the Internet, and not hard to follow.) The chief argument
against it seems to arise from the consequence of MA mentioned above, namely
suppose c = aleph_2, then aleph_1 sized sets have Lebesgue measure zero,
which forces us to Freiling's result again (that c now cannot be aleph_1);
likewise for c = aleph_3, or aleph-{anything allowable}, contradicting
Choice (in the form of the Reals being well-orderable.)
But simply suppose that MA is *not* true in the "full" universe V, only
in sub-universes (just as GCH comes out true in L). That does not downplay
the utility of the axiom any more than the utility of those properties of
groups that fail to hold in the broader category of semigroups is downplayed
in our study of groups. On the affirmative side of things it is worth
noting here that a result of Solovay's shows that there exist models of ZFC
in which CH fails where there are subsets of the Reals of cardinality
aleph_1 which are *not measurable*; that property will suffice to block the
"contradiction" by means of which some have used Freiling against Freiling.
For those who believe the CH has a determinate truth value, Freiling's
method can provide a very compelling argument that the CH is FALSE (as both
Godel and Cohen suspected); the independence results say less about the
meaningfulness of the CH than they say about the limitations of ZFC, gold
standard though it may be for the very important basis it has given the
mathematics community for understanding sets.
Tom Dunion
"From Plato to NATO, history has always had its realists."